College Algebra & Trigonometry, 2018a

1.3. QUADRATIC EQUATIONS 29
1.3 Quadratic Equations
Quadratic equations are equations of the second degree. The solution of quadratic
equations has a long history in mathematics going back several thousand years
to the geometric solutions produced by the Babylonian culture. The Indian mathematician
Brahmagupta used ”rhetorical algebra” (algebra written out in words)
in the 7th century to produce solutions to quadratic equations and Arab mathematicians
of 9th and 10th centuries followed simlar methods. Leonardo of Pisa
(also known as Fibonacci) included information on the Arab approach to solving
quadratic equations in his book Liber Abaci, published in 1202.
The quadratic formula is generally used to solve quadratic equations in standard
form: ax 2 + bx + c =0. The solutions for this are:
x = −b ± √ b 2 − 4ac
2a
Now, the question is - why does this formula give solutions to the standard
quadratic equation? We can proceed as we normally do in solving linear equations
- that is, by getting the x by itself. The only problem here is that instead of
just x, there are also terms involving x 2 . This is where the process of completing
the square comes in handy.
We can begin with the quadratic equation in standard form:
ax 2 + bx + c =0
Just as it is easier to factor a quadratic trinomial if the leading coefficient is 1, this
process of completing the square is also easier if the leading coefficient is 1. So,
next we will divide through on both sides of this equation by a.
ax 2 + bx + c
a
= 0 a
ax 2
a
+ bx a + c a = 0 a
x 2 + b a x + c a =0